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+<HTML>
+<HEAD>
+	<TITLE>Xiangjin Xu - Home Page</TITLE>
+<H1 CLASS="western" ALIGN=CENTER>Personal Home Page of Xiangjin Xu</H1>
+</HEAD>
+<BODY LANG="en-US" DIR="LTR">
+<P STYLE="border-top: none; border-bottom: 1.10pt double #808080; border-left: none; border-right: none; padding-top: 0in; padding-bottom: 0.02in; padding-left: 0in; padding-right: 0in">
+<BR>
+</P>
+<HR>
+<TABLE WIDTH=780 BORDER=1 CELLPADDING=4 CELLSPACING=3>
+	<COL WIDTH=1160>
+	<TR>
+		<TD WIDTH=1000 VALIGN=TOP>
+			<UL>
+				<UL>
+					<!-- <H2 CLASS="western" ALIGN=CENTER><A HREF="CV-updated.pdf"><FONT FACE="Times New Roman, serif"><FONT SIZE=5STYLE="font-size: 18pt">MY
+					CURRICULUM VITAE</FONT></FONT></A></H2>-->
+				</UL>
+			</UL>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+			<H2 CLASS="western" ALIGN=CENTER><FONT FACE="Times New Roman, serif"><FONT SIZE=5 STYLE="font-size: 18pt">RESEARCH INSTERESTS
+			</FONT></FONT>
+			</H2>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+<H2 CLASS="western" ALIGN=LEFT><FONT SIZE=4 STYLE="font-size: 16pt">I. Harmonic Analysis on Manifolds:</FONT></H2>
+			<UL>
+<LI><P><FONT SIZE=4 STYLE="font-size: 14pt">
+Detailed study of the relationship between the growth estimates ( $L^p$ , bilinear, multilinear, and gradient estimates) of the eigenfunctions and the global geometric properties on compact manifolds. Apply the eigenfunction estimates to study the location, distribution and size of nodal sets of eigenfunctions, and to study H\"ormander multiplier problems, Bochner-Riesz means for eigenfunction expansion on compact manifolds.
+</FONT></FONT></P><LI><P><FONT SIZE=4 STYLE="font-size: 14pt">
+Apply the eigenfunction estimates for spectral projectors on manifolds (with or without boundary) to study well-posedness problems for partial differential equations on compact manifolds, including linear or nonlinear wave equations, Schr\"odinger equations, 2D (dissipative) quasi-geostrophic equations, and 2D Euler equations.
+</FONT></FONT></P>
+			</UL>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+<P ALIGN=LEFT><FONT SIZE=4 STYLE="font-size: 16pt"><B>II. Nonlinear differential equations: </B></FONT>
+			</P>
+			<UL>
+<LI><P ALIGN=LEFT><FONT SIZE=4 STYLE="font-size: 14pt">
+Study Li-Yau type sharp differential Harnack inequalities, the heat kernel estimates, and the monotonicity of entropy for linear heat equations and Schr\"odinger operators on Riemannian manifolds with negative Ricci curvature. Study Liouville's Theorems for Schr\"odinger operators on Riemannian manifolds with nonnegative Ricci curvature.
+</FONT></P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 14pt">
+Study gradient estimates for degenerate parabolic equations and Liouville's Theorems, local Aronson-Benilan estimates and entropy formulae for Porous Media Equations and Fast Diffusion Equations.
+</FONT></P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 14pt">
+Study the global uniqueness problems and the boundary stabilization, controllability and observability problems for (linear and nonlinear) parabolic and hyperbolic PDE's on manifolds via Carleman estimates.
+</FONT>
+				</P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 14pt">
+Study the Periodic solutions, subharmonics and homoclinic orbits of Hamiltonian systems.
+</FONT></FONT></P>
+			</UL>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+			<H2 CLASS="western" ALIGN=CENTER><FONT SIZE=4 STYLE="font-size: 16pt"><FONT SIZE=5 STYLE="font-size: 18pt">THESIS
+			</FONT> </FONT>
+			</H2>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+			<OL>
+<LI><P><FONT SIZE=4 STYLE="font-size: 16pt"><FONT SIZE=5><B>Master Thesis:</B></FONT>
+Periodic solutions of Hamiltonian systems and differential systems. Nankai Institute of Mathematics, Tianjin,
+				China, June 1999.
+</FONT></P>
+<LI><P ALIGN=LEFT><FONT SIZE=4 STYLE="font-size: 16pt"><FONT SIZE=5><B>PhD Thesis:</B></FONT>
+Eigenfunction Estimates on Compact Manifolds with Boundary and H\&quot;ormander Multiplier Theorem. Johns Hopkins University, Baltimore, Maryland, May 2004.(<A HREF="thesis.pdf">PDF</A>)</FONT></P>
+			</OL>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+			<H2 CLASS="western" ALIGN=CENTER><FONT SIZE=4 STYLE="font-size: 16pt"><FONT SIZE=5 STYLE="font-size: 18pt">PUBLICATIONS</A>
+			</FONT> </FONT>
+			</H2>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+		<OL>
+                                    <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Subharmonic solutions of a class of non-autonomous Hamiltonian systems. <I>Acta Sci. Nat. Univer. Nankai.</I> Vol. 32, No.2, (1999), pp. 46-50.(In Chinese)</FONT></P>
+                                   <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+Yiming Long, <B>Xiangjin Xu</B>, Periodic solutions for a class of nonautonomous Hamiltonian systems. </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>Nonlinear Anal. Ser. A: Theory Methods, </I></FONT><FONT SIZE=4 STYLE="font-size: 16pt">41 (2000), no. 3-4, 455-463. (<A HREF="http://people.math.binghamton.edu/xxu/Long-Xu.pdf">PDF</A>)</FONT></P>
+                                    <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Homoclinic orbits for first order Hamiltonian systems possessing super-quadratic potentials. </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>Nonlinear Anal. Ser. A: Theory Methods,</I></FONT> <FONT SIZE=4 STYLE="font-size: 16pt">51 (2002), no. 2, 197-214. (<A HREF="http://people.math.binghamton.edu/xxu/Xu-homoclinic.pdf">PDF</A>)</FONT></P>
+                                   <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Periodic solutions for non-autonomous Hamiltonian systems possessing super-quadratic potentials. </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>Nonlinear Anal. Ser. A: Theory Methods,</I></FONT> <FONT SIZE=4 STYLE="font-size: 16pt">51 (2002), no. 6, 941-955. (<A HREF="http://people.math.binghamton.edu/xxu/Xu-periodicsolution.pdf">PDF</A>)</FONT></P>
+                                   <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Subharmonics for first order convex nonautonomous Hamiltonian systems. </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>J. Dynam. Differential Equations</I></FONT> <FONT SIZE=4 STYLE="font-size: 16pt">15 (2003), no. 1, 107-123. (<A HREF="http://people.math.binghamton.edu/xxu/subharmonic-revised.pdf">PDF</A>)</FONT></P>
+                               <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Multiple solutions of super-quadratic second order dynamical systems. Dynamical systems and differential equations (Wilmington, NC, 2002). </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>Discrete Contin. Dyn. Syst.</I></FONT> <FONT SIZE=4 STYLE="font-size: 16pt">2003, suppl., 926-934. (<A HREF="http://people.math.binghamton.edu/xxu/msds.pdf">PDF</A>)</FONT></P>
+                               <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Sub-harmonics of first order Hamiltonian systems and their asymptotic behaviors. Nonlinear differential equations, mechanics and bifurcation (Durham, NC, 2002). </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>Discrete Contin. Dyn. Syst. Ser. B</I></FONT> <FONT SIZE=4 STYLE="font-size: 16pt">3 (2003), no. 4, 643-654. (<A HREF="http://people.math.binghamton.edu/xxu/subharmonic-asym.pdf">PDF</A>)</FONT></P>
+                              <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Homoclinic orbits for first order Hamiltonian systems with convex potentials. </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>Advanced Nonlinear Studies </I></FONT><FONT SIZE=4 STYLE="font-size: 16pt">6 (2006), 399-410. (<A HREF="http://people.math.binghamton.edu/xxu/homoclinic-convex-HS.pdf">PDF</A>)</FONT></P>
+                                 <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, New Proof of H\&quot;ormander Multiplier Theorem on Compact manifolds without boundary. </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>Proc. Amer. Math. Soc. </I></FONT><FONT SIZE=4 STYLE="font-size: 16pt">135 (2007), 1585-1595.(<A HREF="http://www.ams.org/journals/proc/2007-135-05/S0002-9939-07-08687-X/home.html">PDF</A>)</FONT></P>
+                           <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+Roberto Triggiani, <B>Xiangjin Xu</B>, Pointwise Carleman Estimates, Global Uniqueness, Observability, and Stabilization for Schrodinger Equations on Riemannian Manifolds at the  $H^1$ -Level. </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>AMS
+ Contemporary Mathematics</I></FONT><FONT SIZE=4 STYLE="font-size: 16pt">, Volume 426, 2007, 339-404. (<A HREF="http://people.math.binghamton.edu/xxu/RT02-06AMS.pdf">PDF</A>)</FONT></P>
+                        <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Gradient estimates for eigenfunctions of compact manifolds with boundary and the H\&quot;ormander multiplier theorem. </FONT><FONT SIZE=4 STYLE="font-size: 16pt"><I>Forum Mathematicum</I></FONT> <FONT SIZE=4 STYLE="font-size: 16pt">21:3 (May 2009), pp. 455-476. (<A HREF="http://www.degruyter.com/view/j/form.2009.21.issue-3/forum.2009.021/forum.2009.021.xml">PDF</A>)</FONT></P>
+                         <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Eigenfunction estimates for Neumann Laplacian on compact manifolds with boundary and multiplier problems. Proc. Amer. Math. Soc. 139 (2011), 3583-3599.(<A HREF="http://www.ams.org/journals/proc/2011-139-10/S0002-9939-2011-10782-2/home.html">PDF</A>)</FONT></P>
+            <LI><P><A NAME="ddDoi"></A><A NAME="ddJrnl"></A><FONT SIZE=4 STYLE="font-size: 16pt">
+Junfang Li, <B>Xiangjin Xu</B>, Differential Harnack inequalities on Riemannian manifolds I : linear heat equation.Advance in Mathematics, Volume 226, Issue 5, (March, 2011) Pages 4456-4491 <A HREF="http://www.sciencedirect.com/science/article/pii/S0001870810004421">doi:10.1016/j.aim.2010.12.009</A>
+			(<A HREF="http://front.math.ucdavis.edu/0901.3849">arXiv:0901.3849</A>
+			) </FONT>			</P>
+			<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+Liangui Wang, <B>Xiangjin Xu</B>, Hybrid state feedback, robust  $H_{\infty}$  control for a class switched systems with nonlinear uncertainty. </FONT> <FONT SIZE=4 STYLE="font-size: 16pt"> Z. Qian et al.(Eds.):Recent Advances in CSIE 2011,
+<A HREF="http://link.springer.com/chapter/10.1007/978-3-642-25778-0_29">Lecture Notes in Electrical Engineering, Volume 129, 2012, pp 197-202 </A></FONT></P>
+			<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Gradient estimates for  $u_t=\Delta F(u)$  on manifolds and some Liouville-type theorems. Journal of Differential Equation (2011) <A HREF="http://www.sciencedirect.com/science/article/pii/S0022039611003184">doi:10.1016/j.jde.2011.08.004</A>
+			<A HREF="http://front.math.ucdavis.edu/0805.3676">arXiv:0805.3676</A>			</FONT>			</P>
+			<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Upper and lower bounds for normal derivatives of spectral clusters of Dirichlet Laplacian. Journal of Mathematical Analysis and Applications, Volume 387, Issue 1, (March, 2012), Pages 374-383  <A HREF="http://www.sciencedirect.com/science/article/pii/S0022247X11008511">doi:10.1016/j.jmaa.2011.09.003
+			</A>, </FONT><A HREF="http://front.math.ucdavis.edu/1004.2517"><FONT FACE="CMR12"><FONT SIZE=4 STYLE="font-size: 16pt">ArXiv:1004.2517
+			</FONT></FONT></A>			</P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+Huichao Chen, <B>Xiangjin Xu</B>, Power analysis of a left-truncated normal mixture distribution with
+applications in red blood cell velocities. Presentation (by <B>H. Chen</B>) at Joint Statistical Meetings (JSM),
+Montreal, August, 2013.(<A HREF="CX-poweranalysis.pdf"></A>)</FONT></P>
+                      <LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Characterization of Carleson Measures via Spectral Estimates on Compact Manifolds with Boundary. In: Wanduku, D., Zheng, S., Zhou, H., Chen, Z., Sills, A., Agyingi, E. (eds) Applied Mathematical Analysis and Computations I. SGMC 2021. Springer Proceedings in Mathematics & Statistics, vol 471. Springer, Cham. 2024. <A HREF="https://doi.org/10.1007/978-3-031-69706-7_1">https://doi.org/10.1007/978-3-031-69706-7_1</A>(<A HREF="Xu-Carleson.pdf"></A>)</FONT></P>
+			</OL>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+			<UL>
+				<UL>
+					<UL>
+						<UL>
+							<UL>
+								<UL>
+									<P ALIGN=CENTER STYLE="margin-right: 1in; text-decoration: none">
+									<FONT SIZE=5 STYLE="font-size: 18pt"><A HREF="preprints.html"><B>PREPRINTS AND WORK IN PROGRESS</A> </B></FONT>
+									</P>
+								</UL>
+							</UL>
+						</UL>
+					</UL>
+				</UL>
+			</UL>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+		<B><OL></B>
+<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>,  Heat kernel Gaussian bounds on manifolds I: manifolds with non-negative Ricci curvature, 	arXiv:1912.12758 [math.DG] 	(<A HREF="Xu-HeatKernel.pdf"></A>)</FONT></P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Sharp Gradient and Laplacian Estimates for the Heat Kernel on Complete Manifolds with Nonnegative Ricci Curvature, preprint.
+	(<A HREF="Xu-HeatKernel-II.pdf"></A>)</FONT></P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Heat kernel Gaussian bounds on manifolds II: manifolds with negative Ricci curvature, preprint.
+	(<A HREF="Xu-HeatKernel-II.pdf"></A>)</FONT></P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>,  Sharp Hamilton's Gradient and Laplacian Estimates on noncompact manifolds. preprint.
+	(<A HREF="Xu-HeatKernel-II.pdf"></A>)</FONT></P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Differential Harnack inequalities on Riemannian manifolds II: Schr\"odinger operator. (<A HREF="LX-DHI-II.pdf"></A>)</FONT></P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Multiple periodic solutions of super-quadratic Hamiltonian systems with bounded forcing
+terms. (<A HREF="Xu-HS-BF.pdf"></A>)</FONT></P>
+<LI><P><FONT SIZE=4 STYLE="font-size: 16pt">
+<B>Xiangjin Xu</B>, Periodic and subharmonic solutions of Hamiltonian systems possessing "super-quadratic" potentials.  (<A HREF="Xu-HS-SQ.pdf"></A>)</FONT></P>
+			</OL>
+		</TD>
+	</TR>
+	<TR>
+		<TD WIDTH=991 VALIGN=TOP>
+			<UL>
+				<P><FONT SIZE=4 STYLE="font-size: 16pt">My research is
+				partially supported by <B>the NSF Grant <A HREF="http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=0602151">NSF-DMS
+				0602151</A>(June 1 2006-November 30, 2008) and <A HREF="http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=0852507">NSF-DMS-0852507</A>
+				(June 1, 2008-May 31, 2010)</B>, and partially supported by </FONT><FONT FACE="Times New Roman, serif"><FONT SIZE=4 STYLE="font-size: 18pt"><B>Harpur
+				College Grant in Support of Research, Scholarship and Creative
+				Work in Year 2010-2011, 2012-2013, 2018-2019.</B></FONT></FONT></P>
+			</UL>
+		</TD>
+	</TR>
+</TABLE>
+<HR>
+<P>Last updated: 05/01/2015
+</P>
+</BODY>
+</HTML>

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